Question

Find the derivative of the function: f(x)=(5-e^-4x)^8

Answer 1

\(f(x)=(5-e^{-4x})^8\)
Apply the chain rule.

Answer 2

ok i tried but do u know the steps or where i can find the steps

Answer 3

Chain rule: \((f\circ g)'(x)=f'(g(x))\cdot g'(x)\)

Answer 4

In this case \(f(x)=x^8, g(x)=5-e^{-4x}\).

Answer 5

ok im still so lost i tried it but it says my answer is wrong

Answer 6

Okay, let's do it step by step. First, we need to find \(f'(g(x)).\) If \(f(x)=x^8, f'(x)=8x^7.\) So that means that \(f'(g(x))=8(g(x))^7=8(5-e^{-4x})^7.\) Now that we have that term, let's find \(g'(x)\). \(g(x)=5-e^{-4x}\), and it turns out we need to actually apply the chain rule again. In this case, we have \(f(x)=5-e^x, g(x)=-4x\). If we derive \(f\), we get \(f'(x)=-e^x\), so \(f'(g(x))=-e^{g(x)}=-e^{-4x}\). Now, we do \(g'(x)=-4\), so the full term for this application of the chain rule is \(-4\cdot-e^{-4x}=4e^{-4x}\), and if we plug this back into the original application of the chain rule we get \(32e^{-4x}(5-e^{-4x})^7.\)

Answer 7

ok i figured it out thanks